1.8 Combinations of Functions: Composite Functions

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Chapter 1 Functions and Their Graphs

1.8 Combinations of Functions: Composite Functions

What you should learn

? Add, subtract, multiply, and divide functions.

? Find the composition of one function with another function.

? Use combinations and compositions of functions to model and solve real-life problems.

Why you should learn it

Compositions of functions can be used to model and solve real-life problems. For instance, in Exercise 68 on page 92, compositions of functions are used to determine the price of a new hybrid car.

Arithmetic Combinations of Functions

Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to create new functions. For example, the functions given by f x 2x 3 and gx x 2 1 can be combined to form the sum, difference, product, and quotient of f and g.

f x gx 2x 3 x2 1

x 2 2x 4

Sum

f x gx 2x 3 x2 1

x 2 2x 2

Difference

f xgx 2x 3x2 1

2x 3 3x 2 2x 3

Product

f x gx

2x x2

3 1

,

x

?1

Quotient

The domain of an arithmetic combination of functions f and g consists of all real numbers that are common to the domains of f and g. In the case of the quotient fxgx, there is the further restriction that gx 0.

? Jim West/The Image Works

Sum, Difference, Product, and Quotient of Functions

Let f and g be two functions with overlapping domains. Then, for all x common to both domains, the sum, difference, product, and quotient of f and g are defined as follows.

1. Sum:

f gx f x gx

2. Difference: f gx f x gx

3. Product: fgx f x gx

4. Quotient:

f g

x

f x gx

,

gx 0

Example 1 Finding the Sum of Two Functions

Given f x 2x 1 and gx x2 2x 1, find f gx. Solution

f gx f x gx 2x 1 x 2 2x 1 x 2 4x Now try Exercise 5(a).

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Additional Examples

a. Given f x x 5 and gx 3x, find fgx.

Solution

fgx f x gx

x 53x

3x2 15x

b.

Given f x

1 x

and gx

x

x

1,

find gf x.

Solution

gf x gx f x

x

x

11x

x

1

1,

x

0

Section 1.8 Combinations of Functions: Composite Functions

85

Example 2 Finding the Difference of Two Functions

Given f x 2x 1 and gx x2 2x 1, find f gx. Then evaluate the difference when x 2. Solution The difference of f and g is

f gx f x gx 2x 1 x 2 2x 1 x 2 2.

When x 2, the value of this difference is f g2 22 2 2.

Now try Exercise 5(b).

In Examples 1 and 2, both f and g have domains that consist of all real numbers. So, the domains of f g and f g are also the set of all real numbers. Remember that any restrictions on the domains of f and g must be considered when forming the sum, difference, product, or quotient of f and g.

Example 3 Finding the Domains of Quotients of Functions

Find f x and g x for the functions given by

g

f

f x x and gx 4 x 2 .

Then find the domains of fg and gf.

Solution

The quotient of f and g is

f g

x

f x gx

x 4 x 2

and the quotient of g and f is

g f

x

gx f x

4 x 2 .

x

The domain of f is 0, and the domain of g is 2, 2. The intersection of

these domains is 0, 2. So, the domains of f and g are as follows.

g

f

Domain of f : 0, 2 g

Domain of g : 0, 2 f

Note that the domain of fg includes x 0, but not x 2, because x 2 yields a zero in the denominator, whereas the domain of gf includes x 2, but not x 0, because x 0 yields a zero in the denominator.

Now try Exercise 5(d).

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Chapter 1 Functions and Their Graphs

f?g

x

g(x)

g

f

Domain of g Domain of f

FIGURE 1.90

Composition of Functions

Another way of combining two functions is to form the composition of one with the other. For instance, if f x x2 and gx x 1, the composition of f with g is

f gx f x 1

x 12. This composition is denoted as f g and reads as "f composed with g."

f (g(x))

Definition of Composition of Two Functions

The composition of the function f with the function g is

f gx f gx. The domain of f g is the set of all x in the domain of g such that gx is in the domain of f. (See Figure 1.90.)

The following tables of values help illustrate the composition f gx given in Example 4.

x

012 3

gx 4 3 0 5

gx 4 3 0 5 f gx 6 5 2 3

x

012 3

f gx 6 5 2 3

Note that the first two tables can be combined (or "composed") to produce the values given in the third table.

Example 4 Composition of Functions

Given f x x 2 and gx 4 x2, find the following.

a. f gx b. g f x c. g f 2 Solution a. The composition of f with g is as follows.

f gx f gx f 4 x 2

Definition of f g Definition of gx

4 x 2 2

Definition of f x

x2 6

Simplify.

b. The composition of g with f is as follows.

g f x g f x gx 2

Definition of g f Definition of f x

4 x 22

Definition of gx

4 x2 4x 4

Expand.

x2 4x

Simplify.

Note that, in this case, f gx g f x. c. Using the result of part (b), you can write the following.

g f 2 22 42 4 8

Substitute. Simplify.

4

Simplify.

Now try Exercise 31.

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Section 1.8 Combinations of Functions: Composite Functions

87

Te c h n o l o g y

You can use a graphing utility to determine the domain of a composition of functions. For the composition in Example 5, enter the function composition as

y 9 x2 2 9.

You should obtain the graph shown below. Use the trace feature to determine that the x-coordinates of points on the graph extend from 3 to 3. So, the domain of f gx is 3 x 3.

1

-5

5

Example 5 Finding the Domain of a Composite Function

Given f x x2 9 and gx 9 x2, find the composition f gx. Then find the domain of f g. Solution

f gx f gx

f 9 x2 9 x2 2 9

9 x2 9

x2

From this, it might appear that the domain of the composition is the set of all real numbers. This, however is not true. Because the domain of f is the set of all real numbers and the domain of g is 3 x 3, the domain of f g is 3 x 3.

Now try Exercise 35.

- 10

Activities

1. Given f x 3x2 2 and gx 2x, find f g. Answer: f gx 12x2 2

2. Given the functions

f

x

x

1

2

and

gx

x

,

find the composition of f with g.

Then find the domain of the

composition.

Answer:

f

gx

1 x

2.

The

domain of f g is the set of all nonnegative real numbers except x 4.

3. Find two functions f and g such that

f gx hx. (There are many correct answers.)

a. hx 1 3x 1

Answer: f x 1 and x

gx 3x 1

b. hx 2x 34

Answer: f x x 4 and gx 2x 3

In Examples 4 and 5, you formed the composition of two given functions. In calculus, it is also important to be able to identify two functions that make up a given composite function. For instance, the function h given by

hx 3x 53

is the composition of f with g, where f x x3 and gx 3x 5. That is,

hx 3x 53 gx3 f gx.

Basically, to "decompose" a composite function, look for an "inner" function and an "outer" function. In the function h above, gx 3x 5 is the inner function and f x x3 is the outer function.

Example 6 Decomposing a Composite Function

Write

the

function

given

by

hx

x

1

22

as

a

composition

of

two

functions.

Solution

One way to write h as a composition of two functions is to take the inner function to be gx x 2 and the outer function to be

f

x

1 x2

x2.

Then you can write

hx

x

1

22

x

22

f

x

2

f

gx.

Now try Exercise 47.

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Chapter 1 Functions and Their Graphs

Application

Example 7 Bacteria Count

Writing About Mathematics

To expand on this activity, you might consider asking your students to use the tables they created in parts (a) and (b), along with a table of values for x and f x, to demonstrate and explain how the tables can be manipulated to yield tables of values for hx and gx.

The number N of bacteria in a refrigerated food is given by

NT 20T 2 80T 500, 2 T 14

where T is the temperature of the food in degrees Celsius. When the food is removed from refrigeration, the temperature of the food is given by

Tt 4t 2, 0 t 3

where t is the time in hours. (a) Find the composition NTt and interpret its meaning in context. (b) Find the time when the bacterial count reaches 2000.

Solution a. NTt 204t 22 804t 2 500

2016t 2 16t 4 320t 160 500

320t 2 320t 80 320t 160 500

320t 2 420

The composite function NTt represents the number of bacteria in the food as a function of the amount of time the food has been out of refrigeration.

b. The bacterial count will reach 2000 when 320t 2 420 2000. Solve this equation to find that the count will reach 2000 when t 2.2 hours. When you solve this equation, note that the negative value is rejected because it is not in the domain of the composite function.

Now try Exercise 65.

W M RITING ABOUT

ATHEMATICS

Analyzing Arithmetic Combinations of Functions

a. Use the graphs of f and f g in Figure 1.91 to make a table showing the values of gx when x 1, 2, 3, 4, 5, and 6. Explain your reasoning.

b. Use the graphs of f and f h in Figure 1.91 to make a table showing the values of hx when x 1, 2, 3, 4, 5, and 6. Explain your reasoning.

y

y

y

6

f

5 4 3 2 1

x 123456

6

5

f+g

4

3

2

1

x 123456

6

5

f-h

4

3

2

1

x 123456

FIGURE 1.91

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